I have a big problem to solve this system
$\Delta f-hf^2=0$

$|\nabla f|^2+hf^3=0$

where $h$ is a constant, $f$ is a 2-dimensional smooth function, $\Delta f$ is Laplacian of $f$ (i.e. $\Delta f=f_{xx}+f_{yy}$) and $\nabla f$ is the gradient of $f$.

ADD
In first case $f$ is defined on $R^2$
and in second case $f$ is defined on surface $S$ ($f:S \rightarrow (0, \infty)$).
is there a solution?
Thank you for help

MODIFICATION after Igor Khavkine answer:
and if the system is 

$\Delta f-hf^2+cf=0$

$|\nabla f|^2+hf^3=0$

(c is another constant)